Optimal transport (OT), as its name suggests, can be understood as finding an efficient way to `move’ or `transport', the mass from a probability distribution to another distribution.
We borrow notations from the survey book [1]} and focus on one of the canonical OT formulations, one that was proposed in [2]}.
A discrete measure with weights \(a\)  on locations \(u_1, \cdots , u_n\)  would be denoted as \(\alpha = \sum _{1\le i \le n} a_i \delta _{u_i}\) , where notation \(\delta _u\)  stands for a Dirac mass at location \(u\) .
Similarly, for weights \(b\)  on locations \(v_1, \cdots , v_m\)  we have \(\beta = \sum _{1\le j \le m} b_j \delta _{v_j}\) .
A possible way to map a discrete measure \(\alpha \)  onto \(\beta \) , given a cost matrix \(\mathbf {C} \in \mathbb {R}_{+}^{n\times m}\) ,
can be represented with a coupling matrix \(\mathbf {P} \in \mathbb {R}_{+}^{n\times m}\) , where the amount of mass transported from the \(i\) -th location in \(\alpha \)  to \(j\) -th location in \(\beta \)  is stored as \(\mathbf {P}_{i,j}\) .
The set of admissible couplings, \(\mathbf {U}\) , is defined through \(a\)  and \(b\)  as
\(\mathbf {U}(a, b) \mathrel {\overset{\makebox{[}0pt]{\mbox{\normalfont \tiny \sffamily def}}}{=}}\left\lbrace  \mathbf {P} \in \mathbf {R}_{+}^{n\times m}: \sum _j \mathbf {P}_{i,j} = a, \quad \sum _i \mathbf {P}_{i,j} = b \right\rbrace ,\) 
